Steady-state helices of the actin homolog MreB inside bacteria: Dynamics without motors

Within individual bacteria, we combine force-dependent polymerization dynamics of individual MreB protofilaments with an elastic model of protofilament bundles buckled into helical configurations. We use variational techniques and stochastic simulations to relate the pitch of the MreB helix, the total abundance of MreB, and the number of protofilaments. By comparing our simulations with mean-field calculations, we find that stochastic fluctuations are significant. We examine the quasistatic evolution of the helical pitch with cell growth, as well as time scales of helix turnover and de novo establishment. We find that while the body of a polarized MreB helix treadmills toward its slow-growing end, the fast-growing tips of laterally associated protofilaments move toward the opposite fast-growing end of the MreB helix. This offers a possible mechanism for targeted polar localization without cytoplasmic motor proteins.

Figure 1

Five models for the MreB ultrastructure, each composed of polarized protofilaments. (a) Single polarized bundle of protofilaments with lateral interactions that prevent relative slipping within the bundle. As indicated by the star, data in subsequent figures are for this model. (b) Polarized bundle of protofilaments that freely slip relative to each other. (c) Unpolarized bundle of protofilaments, with no lateral slipping. (d) Two antiparallel bundles of polarized protofilaments that slip relative to each other, though protofilaments within a given bundle do not slip. (e) Unpolarized bundle of protofilaments that all freely slip with respect to each other.

Figure 2

Steady-state pitch $p$ vs total molecule number $N_0$ for various filament bulk thicknesses $n$ as predicted by stochastic simulation. The mean-field plot is indistinguishable from this due to the low cytoplasmic concentration. The rectangles represent approximate regions of experimental relevance from [7, 9, 14]. In B. subtilis, if the three isoforms bundle together into a nonslipping triplex structure, the number of monomers will be the sum of each homolog.

Figure 3

Steady-state cytoplasmic concentration $c$ of MreB monomers vs total monomer number $N_0$ from stochastic simulation for $n$ protofilaments, showing $n=1$, 5, 10, 15, 20, and 25. The dotted line represents the critical concentration. The dashed line illustrates the large standard deviation of stochastic fluctuations in steady state, for $n=5$. For all $n$, relative fluctuations were $50\%\pm 5\%$ and absolute fluctuations were within 0.2 nM of those shown for $n=5$.

Figure 4

Maximal steady-state force generation $F$ by a filament bundle with a stochastic $n_{tip}$ as a function of steady-state concentration $c$. Various average bundle thicknesses $n$ are shown. The dashed lines represent the mean-field predicted force-concentration relation [Eq. (B1)] if $n_{tip}=n$ independent tips were all sharing the load. (a) We hold $c$ fixed and allow an arbitrary bundle stiffness. The solid line represents the analytic prediction for $n=2$ with a stochastic $n_{tip}$. The points indicate stochastic simulations, only allowing $n_{tip}$ to vary. Fluctuations in $n_{tip}$ allow a significantly increased force compared to the mean-field results. (b) The points indicate fully stochastic simulations, for the same $n$ as in (a), and the average $c$ is used. A specific bundle elasticity is imposed by forcing $F=F_B$. Fluctuations in $c$ systematically decrease the bundle force compared to (a), and this effect is stronger for smaller $c$.

Figure 5

Pitch $p$ as the cell length $L_c$ grows for various filament bulk thicknesses $n_{bulk}$ as predicted by stochastic simulation, for (a) $N_0=3\times {10}^3{L}_c$, so that an average $3\text{−}\mu \text{m}$ cell contains 9000 MreB monomers, corresponding to regimes toward the low-$N_0$ end of Fig. 2, and (b) $N_0={10}^4{L}_c$, corresponding to the high-$N_0$ end. For most bundle thicknesses, the pitch is effectively constant as the cell elongates. However, for thicker bundles in the low-$N_0$ regime, the helical pitch exhibits a significant dependence on the cell length, expanding as the cell doubles in size.

Figure 6

Schematic of the different possible modes of transport along MreB bundles. Velocities are absolute with respect to the cell in the indicated directions. Bulk treadmilling advects bundles that span the cell length at a speed $v_{tread}$ toward the slow-growing pointed (“$-$”) end. Side associated protofilaments, without constraint forces at their tips, have different polymerization rates and so are not simply advected with the bundle. Their tip velocities are $v_p$ and $v_b$ for pointed and barbed ends, respectively. As discussed in the text, $v_b$ is always opposite the treadmilling direction. Putative motor proteins would probably be polarized and would have a characteristic speed $v_{mot}$. The figure illustrates one polarized protofilament bundle; however, it is possible that two oppositely polarized bundles would exist within a cell—in which case the polarity and velocities of the second bundle should be opposite the first.

Figure 7

Mean translocation times $⟨t_{trans}⟩$ for macromolecule passengers being transported by MreB side protofilaments treadmilling on the side of the main cables vs $N_0$. Constraint forces at the ends of the helix keep $c$ above $c_c$, so that significant axial movement can be seen with respect to bulk treadmilling. Different modes of transport along the cables are depicted schematically in Fig. 6. Barbed ends move antiparallel with respect to bulk advection (a),(c). Pointed ends move in the same direction as bulk advection; however, the slower polymerization rates lead to much longer translocation times (b),(d). Lateral traffic associated with parts of the bulk cables move toward the pointed end at $a_0{\lambda }_{tread}$, indicated in (c) by the dashed line. Any passenger dissociation and reassociation from protofilament tips will increase the translocation times and decrease the effective speeds.